1. EEE 315 - Electrical Properties of Materials
Lecture 2

2. Bravais Lattice
The unit cell of a general 3D lattice is described by 6 numbers
3 distances (a, b, c)
3 angles (, , )
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3. Bravais Lattice
There are 14 distinct 3D lattices which come under 7 Crystal Systems
The BRAVAIS LATTICES (with shapes of unit cells as) :  Cube  (a = b = c,  =  =  = 90)  Square Prism (Tetragonal)  (a = b  c,  =  =  = 90)  Rectangular Prism (Orthorhombic)  (a  b  c,  =  =  = 90)  120 Rhombic Prism (Hexagonal)  (a = b  c,  =  = 90,  = 120)  Parallelepiped (Equilateral, Equiangular)(Trigonal)  (a = b = c,  =  =   90)  Parallelogram Prism (Monoclinic)  (a  b  c,  =  = 90  )  Parallelepiped (general) (Triclinic)  (a  b  c,     )
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4. Bravais Lattice The lattice centering are:
Primitive (P): lattice points on the cell corners only.
Body (I): one additional lattice point at the center of the cell.
Face (F): one additional lattice point at the center of each of the faces of the cell.
Base (A, B or C): one additional lattice point at the center of each of one pair of the cell faces.
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5. Bravais Lattice
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6. Miller Indices
The planes passing through lattice points are called ‘lattice planes’.
The orientation of planes or faces in a crystal can be described in terms of their intercepts on the three axes
Miller introduced a system to designate a plane in a crystal.
He introduced a set of three numbers to specify a plane in a crystal.
This set of three numbers is known as ‘Miller Indices’ of the concerned plane.
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7. Miller Indices Miller indices is defined as the reciprocals of
the intercepts made by the plane on the three
axes.
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8. Miller Indices Procedure for finding Miller Indices
Step 1: Determine the intercepts of the plane
along the axes X,Y and Z in terms of
the lattice constants a,b and c.
Step 2: Determine the reciprocals of these
numbers.
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9. Miller Indices Step 3: Find the least common denominator (lcd)
and multiply each by this lcd.
Step 4:The result is written in parenthesis.
This is called the `Miller Indices’ of the plane in the form
(h k l).
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10. Miller Indices ILLUSTRATION PLANES IN A CRYSTAL
Plane ABC has intercepts of 2 units along X-axis, 3
units along Y-axis and 2 units along Z-axis.
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11. Miller Indices ILLUSTRATION DETERMINATION OF ‘MILLER INDICES’
Step 1:The intercepts are 2,3 and 2 on the three axes.
Step 2:The reciprocals are 1/2, 1/3 and 1/2.
Step 3:The least common denominator is ‘6’.
Multiplying each reciprocal by lcd, we get, 3,2 and 3.
Step 4:Hence Miller indices for the plane ABC is (3 2 3)
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12. Miller Indices Example
In this plane, the intercept along X axis is 1 unit.
The plane is parallel to Y and Z axes. So, the intercepts along Y and Z axes are ‘’.
Now the intercepts are 1,  and .
The reciprocals of the intercepts are = 1/1, 1/ and 1/.
Therefore the Miller indices for the above plane is (1 0 0).
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13. Miller Indices
SOME IMPORTANT PLANES
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14. Miller Indices Problem # 1
A certain crystal has lattice parameters of 4.24, 10 and 3.66 Å on X, Y, Z axes respectively. Determine the Miller indices of a plane having intercepts of 2.12, 10 and 1.83 Å on the X, Y and Z axes.
Lattice parameters are = 4.24, 10 and 3.66 Å
The intercepts of the given plane = 2.12, 10 and 1.83 Å
i.e. The intercepts are, 0.5, 1 and 0.5.
Step 1: The Intercepts are 1/2, 1 and 1/2.
Step 2: The reciprocals are 2, 1 and 2.
Step 3: The least common denominator is 2.
Step 4: Multiplying the lcd by each reciprocal we get, 4, 2 and 4.
Step 5: By writing them in parenthesis we get (4 2 4)
Therefore the Miller indices of the given plane is (4 2 4) or (2 1 2).
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15. Miller Indices Problem # 2
Calculate the miller indices for the plane with intercepts 2a, - 3b and 4c the along the crystallographic axes.
The intercepts are 2, - 3 and 4
Step 1: The intercepts are 2, -3 and 4 along the 3 axes
Step 2: The reciprocals are 1/2, -1/3, 1/4
Step 3: The least common denominator is 12.
Multiplying each reciprocal by lcd, we get and 3
Step 4: Hence the Miller indices for the plane is
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16. Classical Theory of Electrical and Thermal Conduction
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17. Electrical & Thermal Conduction
Electrical conduction: Motion of charges (conduction electrons) in a material under the influence of an applied electric field
Thermal conduction: Conduction of thermal energy from higher to lower temperature regions in a material
Thermal conduction involves carrying of energy by conduction electrons.
Good electrical conductors are also good thermal conductors!
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18. Next Week!
QUIZ!
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